SLAC-PUB-4679 August 1989 (4 CROSS-PLANE COUPLING AND ITS EFFECT ON PROJECTED EMITTANCE’ Karl L. Brown and Roger V. Servranckx Stanford Linear Accelerator Center Stanford University, Stanford, CA 94309 ABSTRACT We have developed a general first-order theory of coupled motion between the transverse phase planes in single-pass static magnetic beam transport systems. The results are expressed in terms of the projected emittances in the z and y transverse planes at any point s along the optical axis of the system. Submitted to Particle Accelerators *Work supported by the Department of Energy, contract DE-ACXK3-76SF00515.
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CROSS-PLANE COUPLING AND ITS EFFECT ON PROJECTED EMITTANCE ... · All Bll Cl1 Dll + = A21 B21 c21 D21 0 7 All B12 Cl1 D12 -t = A21 B22 c21 022 0 9 (32) A12 Bll Cl2 Dll + = A22 B21
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SLAC-PUB-4679 August 1989
(4
CROSS-PLANE COUPLING AND ITS EFFECT
ON PROJECTED EMITTANCE’
Karl L. Brown and Roger V. Servranckx
Stanford Linear Accelerator Center
Stanford University, Stanford, CA 94309
ABSTRACT
We have developed a general first-order theory of coupled motion between the
transverse phase planes in single-pass static magnetic beam transport systems. The
results are expressed in terms of the projected emittances in the z and y transverse
planes at any point s along the optical axis of the system.
Submitted to Particle Accelerators
*Work supported by the Department of Energy, contract DE-ACXK3-76SF00515.
I. INTRODUCTION
One of the possible consequences of coupled motion in a beam transport sys-
tem is an increase in the emittance projected onto the transverse planes at points
downstream. To evaluate the magnitude of this effect we use the symplectic condi-
tion imposed upon the system by Hamiltonian mechanics to minimize the number
of free parameters needed to describe the coupled motion. We then calculate a
four-dimensional monoenergetic beam envelope (a matrix) at any position s along
the optic axis of the transport line and from this determine the emittance projected
onto the I(: and y transverse phase planes. The results are expressed in terms of
a 4x4 linear transformation matrix R and its 2x2 submatrices A, B, C, D. An
important result is that if the determinant of any one of the 2x2 submatrices be-
comes negative, then the projected emittance can become large, depending upon
the magnitudes of the absolute values of the submatrix determinants.
The calculations are applicable to any single-pass charged particle beam trans-
port system constructed from static-magnetic optical elements.
II. NOTATION
For this report we use the definitions and notation of the TRANSPORT1j2 pro-
gram where the monoenergetic linear transformation in a beam transport system
is expressed in the following matrix form:
(1)
-
R is a 4 x4 linear transformation matrix and, to first-order, the coordinates x, x’ ,
y, and y’ form a set of canonical variables.
2
i For static magnetic systems, Liouville’s theorem of phase space conservation
requires that det R = 1. If R has the form
R= (2)
where A and D are 2 x2 submatrices, then the x- and y-plane optics are decoupled
and are independent of each other, in which case
detA = detD = 1 . (3)
Thus, six independent parameters are needed to determine the uncoupled R matrix,
three for the x plane and three for the y plane.
In the more general case, where the motion between the x and y phase planes
is coupled, the R-matrix has the form:
R= (4)
where R is a 4x4 matrix and A, B, C and D are 2x2 submatrices. B is a matrix
coupling the y-plane to the x-plane and C is a matrix coupling the x-plane to the
y-plane.
As will be shown in Section III, there are six independent symplectic conditions
imposed upon R from Hamiltonian mechanics. Therefore there are (sixteen minus
six = ) ten independent parameters needed to uniquely determine the matrix
elements of the coupled R matrix.
-
III. THE SYMPLECTIC CONDITION AND ITS CONSEQUENCES
The role of the symplectic condition for coupled motion in particle accelerators
has been previously addressed by Courant and Snyder3 and Teng5 among others.
For our purpose we begin with a derivation of the symplectic condition and develop
from this the implications to our particular problem.
Let us consider motion in three dimensional space and denote the coordinates
by Q = (~1, qz,q3). As is well known, the motion of particles can be described by
the Hamiltonian equations
dH ii=ap; ,
dH Pi=-% ) (5)
where pi are the conjugate momenta of the variables qi, and H is the Hamiltonian
describing the system.
The transformation & = Q(q, p, t), P = P(q, p, t) is canonical when there exists
a function K(Q, P, t) ( a new Hamiltonian) such that the variables Q, P satisfy the
equations:
Qi=~ , pi=-?5 . I 8Qi (6)
According to Hamiltonian mechanics the functions Q and P satisfy the follow-
ing conditions:
[Qi,Qjl=O 7 [pi, pj] = 0 7 [Qi, Pi] = 6.. ,3 )
where [F, G] is th e P oisson bracket of F, G defined as:
wl=~(gg-gg) * j 3
4
8Pi dPk a?% 8Pk ----- = &jO dPj0 aPj0 dqj0
o 7
Given initial conditions for the variables qi and the momenta pi, a solution exists
which can be written in the form:
Qi = Qi(qiO,piO, t) 7 Pi = Pi(QiO,piO, t) - (7)
This transformation, from the initial conditions to the values at time t, is a canon-
ical transformation. Thus the functions qi and pi satisfy the relations:
[Qi7 qjl = 0 7 [pi,Pjl =O 7 [Qi,Pjl = bj 7 -
or explicitly:
(8)
Let us now consider the six-dimensional vector
u = hPl~Qa,Pa,Qa,Pa)
and the Jacobian of the transformation (7) defined as:
M=
Then the conditions (8) can be expressed in terms of the matrix A4 as follows:
iizs% l = s (9)
where S is the matrix:
- and M is the transpose of M. Equation (9) is called the symplectic condition. All
solutions to Hamiltonian equations must satisfy this condition. Conversely, if a
transform (7) satisfies the conditions (9), then there exists locally a Hamiltonian
and a set of associated Hamiltonian equations to which (7) is a solution.
‘0 1 0 0 0 0
-10 0 0 0 0
000100
0 0 -10 0 0
000001
.o 0 0 0 -1 0
As shown in Appendix B, to first-order, the coordinates x, IC’, y, and y’ form
a set of canonical variables. Therefore the symplectic condition also applies to the
4x4 R matrix, defined in Eq. (l), as follows:
iiSR = S,
where E means the transpose of R and
S is now a 4x4 symplectic matrix defined as
s=
0 1 0 0
-1 0 0 0
0 0 0 1
0 0 -1 0
.
4x4
(10)
(11)
(12)
-
Note that SS = -I (where I is the unity matrix).
6
We now define the kympkectic conjugate, ?i’, of the 2x2 submatrix, A, as:
XC-- [-y ;) Ji (-Y ;] = (-Z;; -y ; (13)
from. which it follows that
Ax = ZA = detA 0
0 I detA * (14)
Similarly, the symplectic conjugate of the 4x4 R-matrix is defined as:
77 = -sky = ?i:E i 1 B D 4x4 - Such that
(15)
(16) RR = -SkS’R = -SS = I .
As a consequence, it follows that:
j&R-l . (17)
We now expand the matrix products RR and XR in terms of their 2x2 sub-
matrices.
RX= [; I] [; ;] , (18)
from which
-
Az+BB RR =
Cz+ DB
7
From Eqs. (14)and (19), it follows that
detA+detB 0 I I 1
ZA+BB= 0 detA+detB 2x2 = 0
and
1 cC+i?D=
detC+detD 0
= 0 detC+detD
1 2x2
I 0
0 1 (21) 1 2x2
0
1 1 2x2
(22)
Similar results may be obtained from Eq. (20).
From Eq. (19) we conclude that the symplectic condition imposes the following
constraints upon the coupled R-matrix.
detA+detB = 1, detC+detD = 1,
Ac+BD=O, Cz+DB=O;
and from Eq. (20) we have the alternate form
detB+detD = 1, detA+detC = 1,
Z~+zli~=o, BA+??C=O.
It can be shown that Eqs. (23) and (24) are two different, but equivalent, ways
of expressing the symplectic constraint. We will use both forms in this report,
-
8
but first we show that the set of Eqs. (24) re p resent only six independent condi-
tions. The first two are:
detA+detC = 1 ,
(25) det B+ det D = 1 .
Now we show that the last of Eqs. (24) p ro d uces four additional conditions.
Given:
BA+i% = 0 , (26)
BA =
BA =
where we define
A11 &2 A12 &2
A21 B22 A22 B22
A11 &I A12 &I - - A21 B21 A22 B21
41 A21
f?iI = det [ ::: ::I] . . .
Similarly,
(27)
, (28)
2x2
(29)
DC = t -;;I -;;; ] ( ;; :;I] ) (30)
9
or / /
Cl1 Cl1 012 012
c21 c21 022 022 DC = DC =
Cl1 Cl1 Dll Dll - - - -
c21 c21 D21 D21 \ \
Cl2
c22
Cl2
c22
\
D12
022
Dll
Da1 J 2x2
(31)
Adding Eqs. (28) and (31), we conclude that Eq. (26) is equivalent to the
following set of four equations:
All Bll Cl1 Dll + =
A21 B21 c21 D21 0 7
All B12 Cl1 D12 -t =
A21 B22 c21 022 0 9
(32)
A12 Bll Cl2 Dll + =
A22 B21 c22 D21 0 7
42 B12 Cl2 012
A22 B22 +c22 = 0 .
D22
It can be shown that expanding the matrix equation
F~B+CD=O,
from Eqs. (24) yields the same set of four Eqs. (32). Thus there are only six
independent relations imposed by the symplectic condition. These results may be
summarized as follows:
10
-
Define the “sum of determinants” of any two columns of the matrix R as the
sum of the two determinants formed by rows 1 and 2 and rows 3 and 4. There
are six such combinations. The above six symplectic conditions represent all of
the combinations of the columns (;,j). If (;,j) = (1,2) or (3,4), then the sum of
the determinants is equal to 1, which is equivalent to Eqs. (25). For the remaining
combinations of columns, such as (1,3), th e sum of the determinants is equal to
zero, which is equivalent to Eqs. (32). It is readily deduced from Eqs. (25) and
(32) that if all of the matrix elements of B are identically equal to zero, then the
same is true for the matrix elements of C.
If one interchanges the words, rows, and columns in the above paragraph, then
the resulting statement applies to the set of Eqs. (23). From this one can show that
Eqs. (23) and (24) are equivalent ways of expressing the six symplectic constraints.
IV. CALCULATION OF THE PROJECTED BEAM EMITTANCE
We are now in a position to define and calculate the projected emittances in
the z and y transverse planes of a beam as it passes through a coupled system.
Let 0 be a 4x4 symmetric, positive definite, beam envelope mat&r as defined
and used in TRANSPORT.2
UX t CT= I 1 LTy '
(33)
-
where gx and gY are 2x2 symmetric, positive definite, matrices representing the z
and y projections of the beam, and the 2x2 matrix t describes the 2-y coupling
present in the beam.
11
. The relation between: the beam matrices at a position 0 and at a position 1 is